The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
revconsapp, deeprevapp

They will be analysed ascendingly in the following order:
revconsapp < deeprevapp

(6) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

Generator Equations:
gen_C:V:N4_0(0) ⇔ V(hole_a2_0)
gen_C:V:N4_0(+(x, 1)) ⇔ C(V(hole_a2_0), gen_C:V:N4_0(x))

The following defined symbols remain to be analysed:
revconsapp, deeprevapp

They will be analysed ascendingly in the following order:
revconsapp < deeprevapp

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
revconsapp(gen_C:V:N4_0(0), gen_C:V:N4_0(b)) →_R^Ω(1)
gen_C:V:N4_0(b)

Induction Step:
revconsapp(gen_C:V:N4_0(+(n6_0, 1)), gen_C:V:N4_0(b)) →_R^Ω(1)
revconsapp(gen_C:V:N4_0(n6_0), C(V(hole_a2_0), gen_C:V:N4_0(b))) →_IH
gen_C:V:N4_0(+(+(b, 1), c7_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_C:V:N4_0(0) ⇔ V(hole_a2_0)
gen_C:V:N4_0(+(x, 1)) ⇔ C(V(hole_a2_0), gen_C:V:N4_0(x))

The following defined symbols remain to be analysed:
deeprevapp

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
deeprevapp(gen_C:V:N4_0(n958_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n958_0, b)), rt ∈ Ω(1 + b + n958₀)

Induction Base:
deeprevapp(gen_C:V:N4_0(0), gen_C:V:N4_0(b)) →_R^Ω(1)
revconsapp(gen_C:V:N4_0(b), V(hole_a2_0)) →_L^{Ω(1 + b)}
gen_C:V:N4_0(+(b, 0))

Induction Step:
deeprevapp(gen_C:V:N4_0(+(n958_0, 1)), gen_C:V:N4_0(b)) →_R^Ω(1)
deeprevapp(gen_C:V:N4_0(n958_0), C(V(hole_a2_0), gen_C:V:N4_0(b))) →_IH
gen_C:V:N4_0(+(+(b, 1), c959_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
deeprevapp(gen_C:V:N4_0(n958_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n958_0, b)), rt ∈ Ω(1 + b + n958₀)

Generator Equations:
gen_C:V:N4_0(0) ⇔ V(hole_a2_0)
gen_C:V:N4_0(+(x, 1)) ⇔ C(V(hole_a2_0), gen_C:V:N4_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
deeprevapp(gen_C:V:N4_0(n958_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n958_0, b)), rt ∈ Ω(1 + b + n958₀)

Generator Equations:
gen_C:V:N4_0(0) ⇔ V(hole_a2_0)
gen_C:V:N4_0(+(x, 1)) ⇔ C(V(hole_a2_0), gen_C:V:N4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: a → C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_a2_0 :: a
hole_False:True3_0 :: False:True
gen_C:V:N4_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_C:V:N4_0(0) ⇔ V(hole_a2_0)
gen_C:V:N4_0(+(x, 1)) ⇔ C(V(hole_a2_0), gen_C:V:N4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N4_0(n6_0), gen_C:V:N4_0(b)) → gen_C:V:N4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)